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Scientia Magna Vol. 2, No. 3, 2006 ISSN 1556-6706 SCIENTIA MAGNA Edited by Department of Mathematics Northwest University, P. R. China HIGH AMERICAN PRESS Vol. 2, No. 3, 2006 ISSN 1556-6706 SCIENTIA MAGNA Edited by Department of Mathematics Northwest University Xi’an, Shaanxi, P.R.China High American Press Scientia Magna is published annually in 400-500 pages per volume and 1,000 copies. It is also available in microfilm format and can be ordered (online too) from: Books on Demand ProQuest Information & Learning 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Michigan 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) URL: http://wwwlib.umi.com/bod/ Scientia Magna is a referred journal: reviewed, indexed, cited by the following journals: "Zentralblatt Für Mathematik" (Germany), "Referativnyi Zhurnal" and "Matematika" (Academia Nauk, Russia), "Mathematical Reviews" (USA), "Computing Review" (USA), Institute for Scientific Information (PA, USA), "Library of Congress Subject Headings" (USA). Printed in the United States of America Price: US$ 69.95 i Information for Authors Papers in electronic form are accepted. They can be e-mailed in Microsoft Word XP (or lower), WordPerfect 7.0 (or lower), LaTeX and PDF 6.0 or lower. The submitted manuscripts may be in the format of remarks, conjectures, solved/unsolved or open new proposed problems, notes, articles, miscellaneous, etc. They must be original work and camera ready [typewritten/computerized, format: 8.5 x 11 inches (21,6 x 28 cm)]. They are not returned, hence we advise the authors to keep a copy. The title of the paper should be writing with capital letters. The author's name has to apply in the middle of the line, near the title. References should be mentioned in the text by a number in square brackets and should be listed alphabetically. Current address followed by e-mail address should apply at the end of the paper, after the references. The paper should have at the beginning an abstract, followed by the keywords. All manuscripts are subject to anonymous review by three independent reviewers. Every letter will be answered. Each author will receive a free copy of the journal. ii Contributing to Scientia Magna Authors of papers in science (mathematics, physics, philosophy, psychology, sociology, linguistics) should submit manuscripts, by email, to the Editor-in-Chief: Prof. Wenpeng Zhang Department of Mathematics Northwest University Xi’an, Shaanxi, P.R.China E-mail: [email protected] Or anyone of the members of Editorial Board: Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, IIT Madras, Chennai - 600 036, Tamil Nadu, India. Dr. Larissa Borissova and Dmitri Rabounski, Sirenevi boulevard 69-1-65, Moscow 105484, Russia. Dr. Huaning Liu, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China. E-mail: [email protected] Prof. Yuan Yi, Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi, P.R.China. E-mail: [email protected] Dr. Zhefeng Xu, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China. E-mail: [email protected]; [email protected] Dr. Tianping Zhang, College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, P.R.China. E-mail: [email protected] iii Contents A. Majumdar : A note on the Pseudo-Smarandache function 1 S. Gupta : Primes in the Smarandache deconstructive sequence 26 S. Zhang and C. Chen : Recursion formulae for Riemann zeta function and Dirichlet series 31 A. Muktibodh : Smarandache semiquasi near-rings 41 J. S¶andor: On exponentially harmonic numbers 44 T. Ja¶³y¶eo.l¶a: Parastrophic invariance of Smarandache quasigroups 48 M. Karama : Perfect powers in Smarandache n-expressions 54 T. Ja¶³y¶eo.l¶a: Palindromic permutations and generalized Smarandache palindromic permutations 65 J. S¶andor: On certain inequalities involving the Smarandache function 78 A. Muktibodh : Sequences of pentagonal numbers 81 J. Gao : On the additive analogues of the simple function 88 W. Zhu : On the primitive numbers of power p 92 A. Vyawahare : Smarandache sums of products 96 N. Yuan : On the solutions of an equation involving the Smarandache dual function 104 H. Zhou : An in¯nite series involving the Smarandache power function SP (n) 109 iv Scientia Magna Vol. 2 (2006), No. 3, 1-25 A note on the Pseudo-Smarandache function A.A.K. Majumdar1 Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh Abstract This paper gives some results and observations related to the Pseudo-Smarandache function Z(n). Some explicit expressions of Z(n) for some particular cases of n are also given. Keywords The Pseudo-Smarandache function, Smarandache perfect square, equivalent. x1. Introduction The Pseudo-Smarandache function Z(n), introduced by Kashihara [1], is as follows : De¯nition 1.1. For any integer n ¸ 1, Z(n) is the smallest positive integer m such that 1 + 2 + ¢ ¢ ¢ + m is divisible by n. Thus, ½ ¾ m(m + 1) Z(n) = min m : m 2 N: n j : (1:1) 2 As has been pointed out by Ibstedt [2], an equivalent de¯nition of Z(n) is De¯nition 1.2. p Z(n) = minfk : k 2 N: 1 + 8kn is a perfect squareg: Kashihara [1] and Ibstedt [2] studied some of the properties satis¯ed by Z(n). Their ¯ndings are summarized in the following lemmas: Lemma 1.1. For any m 2 N, Z(n) ¸ 1. Moreover, Z(n) = 1 if and only if n = 1, and Z(n) = 2 if and only if n = 3. Lemma 1.2. For any prime p ¸ 3, Z(p) = p ¡ 1. Lemma 1.3. For any prime p ¸ 3 and any k 2 N, Z(pk) = pk ¡ 1. Lemma 1.4. For any k 2 N, Z(2k) = 2k+1 ¡ 1. Lemma 1.5. For any composite number n ¸ 4, Z(n) ¸ maxfZ(N): N j ng. In this paper, we give some results related to the Pseudo-Smarandache function Z(n). In x2, we present the main results of this paper. Simple explicit expressions for Z(n) are available for particular cases of n. In Theorems 2:1 ¡ 2:11, we give the expressions for Z(2p), Z(3p), Z(2p2), Z(3p3), Z(2pk), Z(3pk), Z(4p), Z(5p), Z(6p), Z(7p) and Z(11p), where p is a prime and k(¸ 3) is an integer. Ibstedt [2] gives an expression for Z(pq) where p and q are distinct primes. We give an alternative expressions for Z(pq), which is more e±cient from the computational point of view. This is given in Theorem 2:12, whose proof shows that the solution of Z(pq) involves the solution of two Diophantine equations. Some particular cases of Theorem 1On Sabbatical leave from: Ritsumeikan Asia-Paci¯c University, 1-1 Jumonjibaru, Beppu-shi, Oita-ken, Japan. 2 A.A.K. Majumdar No. 3 2:12 are given in Corollaries 2:1 ¡ 2:16. We conclude this paper with some observations about the properties of Z(n), given in four Remarks in the last x3. x2. Main Results We ¯rst state and prove the following results. k(k+1) Lemma 2.1. Let n = 2 for some k 2 N. Then, Z(n) = k. Proof. Noting that k(k + 1) = m(m + 1) if and only if k = m, the result follows. The following lemma gives lower and upper bounds of Z(n). Lemma 2.2. 3 · n · 2n ¡ 1 for all n ¸ 4. m(m+1) Proof. Letting f(m) = 2 ; m 2 N, see that f(m) is strictly increasing in m with f(2) = 3. Thus, Z(n) = 2 if and only if n = 3. This, together with Lemma 1:1, gives the lower bound of Z(n) for n ¸ 4. Again, since n j f(2n ¡ 1), it follows that Z(n) cannot be greater than 2n ¡ 1. Since Z(n) = 2n ¡ 1 if n = 2k for some k 2 N, it follows that the upper bound of Z(n) in Lemma 2:2 cannot be improved further. However, the lower bound of Z(n) can be improved. For example, since f(4) = 10, it follows that Z(n) ¸ 5 for all n ¸ 11. A better lower bound of Z(n) is given in Lemma 1:5 for the case when n is a composite number. In Theorems 2:1 ¡ 2:4, we give expressions for Z(2p);Z(3p);Z(2p2) and Z(3p2) where p ¸ 5 is a prime. To prove the theorems, we need the following results. Lemma 2.3. Let p be a prime. Let an integer n(¸ p) be divisible by pk for some integer k(¸ 1). Then, pk does not divide n + 1 (and n ¡ 1). Lemma 2.4. 6 j n(n + 1)(n + 2) for any n 2 N. In particular, 6 j (p2 ¡ 1) for any prime p ¸ 5. Proof. The ¯rst part is a well-known result. In particular, for any prime p ¸ 5, 6 j (p ¡ 1)p(p + 1). But since p(¸ 5) is not divisible by 6, it follows that 6 j (p ¡ 1)(p + 1). Theorem 2.1. If p ¸ 5 is a prime, then 8 < p ¡ 1; if 4 j (p ¡ 1); Z(2p) = : p; if 4 j (p + 1): Proof. ½ ¾ ½ ¾ m(m + 1) m(m + 1) Z(2p) = min m : 2p j = min m : p j : (1) 2 4 If p j m(m + 1), then p must divide either m or m + 1, but not both (by Lemma 2.3). Thus, the minimum m in (1) may be taken as p ¡ 1 or p depending on whether p ¡ 1 or p + 1 respectively is divisible by 4.
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